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Author: Ed Nelson
Department of Sociology M/S SS97
California State University, Fresno
Fresno, CA 93740
Email: [email protected]
Note to the Instructor: This exercise uses the 2014 General Social Survey (GSS) and SDA to
explore measures of skewness and kurtosis. SDA (Survey Documentation and Analysis) is an
online statistical package written by the Survey Methods Program at UC Berkeley and is
available without cost wherever one has an internet connection. The 2014 Cumulative Data File
(1972 to 2014) is also available without cost by clicking here. For this exercise we will only be
using the 2014 General Social Survey. A weight variable is automatically applied to the data set
so it better represents the population from which the sample was selected. You have permission
to use this exercise and to revise it to fit your needs. Please send a copy of any revision to the
author. Included with this exercise (as separate files) are more detailed notes to the instructors
and the exercise itself. Please contact the author for additional information.
I’m attaching the following files.
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Extended notes for instructors (MS Word; .docx format).
This page (MS Word; .docx format).
Goals of Exercise
The goal of this exercise is to explore measures of skewness and kurtosis. The exercise also
gives you practice in using FREQUENCIES in SDA.
Part I – Measures of Skewness
A normal distribution is a unimodal (i.e., single peak) distribution that is perfectly
symmetrical. In a normal distribution the mean, median, and mode are all equal. Here’s a graph
showing what a normal distribution looks like.
The horizontal axis is marked off in terms of standard scores where a standard score tells us how
many standard deviations a value is from the mean of the normal distribution. So a standard
score of +1 is one standard deviation above the mean and a standard score of -1 is one standard
deviation below the mean. The percents tell us the percent of cases that you would expect
between the mean and a particular standard score if the distribution was perfectly normal. You
would expect to find approximately 34% of the cases between the mean and a standard score of
+1 or -1. In a normal distribution, the mean, median, and mode are all equal and are at the center
of the distribution. So the mean always has a standard score of zero.
Skewness measures the deviation of a particular distribution from this symmetrical pattern. In a
skewed distribution one side has longer or fatter tails than the other side. If the longer tail is to
the left, then it is called a negatively skewed distribution. If the longer tail is to the right, then it
is called a positively skewed distribution. One way to remember this is to recall that any value to
the left of zero is negative and any value to the right of zero is positive. Here are graphs of
positively and negatively skewed distributions compared to a normal distribution.
We’re going to use the General Social Survey (GSS) for this exercise. The GSS is a national
probability sample of adults in the United States conducted by the National Opinion Research
Center (NORC). The GSS started in 1972 and has been an annual or biannual survey ever since.
For this exercise we’re going to use the 2014 GSS. To access the GSS cumulative data file in
SDA format click here. The cumulative data file contains all the data from each GSS survey
conducted from 1972 through 2014. We want to use only the data that was collected in
2014. To select out the 2014 data, enter year(2014) in the Selection Filter(s) box. Your screen
should look like Figure 3-1. This tells SDA to select out the 2014 data from the cumulative file.
Figure 3-1
Notice that a weight variable has already been entered in the WEIGHT box. This will weight the
data so the sample better represents the population from which the sample was selected.
The GSS is an example of a social survey. The investigators selected a sample from the
population of all adults in the United States. This particular survey was conducted in 2014 and is
a relatively large sample of approximately 2,500 adults. In a survey we ask respondents
questions and use their answers as data for our analysis. The answers to these questions are used
as measures of various concepts. In the language of survey research these measures are typically
referred to as variables. Often we want to describe respondents in terms of social characteristics
such as marital status, education, and age. These are all variables in the GSS.
Run FREQUENCIES in SDA for the variables age and sibs. To run the frequency distributions,
enter the variable names, age and sibs, in the ROW box. Your screen should like Figure 3-2.
Separate the variable names by either a space or a comma. Notice that the SELECTION
FILTER(S) box and the WEIGHT box are both filled in.
Figure 3-2
Once you have selected these variables click on the arrow next to OUTPUT OPTIONS and
check the box for SUMMARY STATISTICS. Then click on CHART OPTIONS and click the
arrow next to TYPE OF CHART. Select BAR CHART and now click on RUN THE TABLE at
the bottom. SDA will compute the mean and median (plus other statistics) for each variable
along with the bar chart.
Notice that the mean is larger than the median for both variables. This means that the
distribution is positively skewed. But also notice that the mean for sibs is quite a bit larger than
the median in a relative sense than is the case for age. This suggests that the distribution for sibs
is the more skewed of the two variables. Look at the bar charts and you’ll see the same
thing. Both variables are positively skewed but sibs is the more skewed variable. Now look at
the skewness values — 1.72 for sibs and .24 for age. The larger the skewness value, the more
skewed the distribution. Positive skewness values indicate a positive skew and negative values
indicate a negative skew. There are various rules of thumb suggested for what constitutes a lot
of skew but for our purposes we’ll just say that the larger the value, the more the skewness and
the sign of the value indicates the direction of the skew.
Run FREQUENCIES for the following variables. Tell SDA to give you the bar chart along with
the summary statistics. Write a paragraph for each variable explaining what these statistics tell
you about the skewness of the variables.
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hrsrelax
tvhours
Part II – Measures of Kurtosis
Kurtosis refers to the flatness or peakness of a distribution relative to that of a normal
distribution. Distributions that are flatter than a normal distribution are called platykurtic and
distributions that are more peaked are called leptokurtic.
SDA will compute a kurtosis measure. Negative values indicate a platykurtic distribution and
positive values indicate a leptokurtic distribution. The larger the kurtosis value, the more peaked
or flat the distribution is.
Look back at the output for age and sibs. For age the kurtosis value was -.80 indicating a flatter
distribution and for sibs kurtosis was 4.39 indicating a more peaked distribution. To see this
visually look at your bar charts.
Run FREQUENCIES for the following variables. Tell SDA to give you the summary statistics
and a bar chart. Write a paragraph for each variable explaining what these statistics tell you
about the kurtosis of the variables.
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maeduc
paeduc
sexfreq